L(s) = 1 | − 7-s + 13-s − 3·19-s + 4·23-s − 4·29-s + 7·31-s − 6·37-s − 6·41-s − 9·43-s + 6·47-s − 6·49-s − 2·53-s + 10·59-s − 61-s + 3·67-s − 14·71-s + 10·73-s − 8·79-s + 18·83-s − 91-s − 3·97-s + 6·101-s − 8·103-s + 2·107-s − 15·109-s + 12·113-s + ⋯ |
L(s) = 1 | − 0.377·7-s + 0.277·13-s − 0.688·19-s + 0.834·23-s − 0.742·29-s + 1.25·31-s − 0.986·37-s − 0.937·41-s − 1.37·43-s + 0.875·47-s − 6/7·49-s − 0.274·53-s + 1.30·59-s − 0.128·61-s + 0.366·67-s − 1.66·71-s + 1.17·73-s − 0.900·79-s + 1.97·83-s − 0.104·91-s − 0.304·97-s + 0.597·101-s − 0.788·103-s + 0.193·107-s − 1.43·109-s + 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55450369771688035071714991592, −6.73472876033950905183980088011, −6.37095247036193255421476502980, −5.39830694538726179521354814029, −4.80234809036820990737382481889, −3.86244208296568369791086854386, −3.21026016842139130510207246795, −2.28967252494869327056409312340, −1.28329535957897760021317649668, 0,
1.28329535957897760021317649668, 2.28967252494869327056409312340, 3.21026016842139130510207246795, 3.86244208296568369791086854386, 4.80234809036820990737382481889, 5.39830694538726179521354814029, 6.37095247036193255421476502980, 6.73472876033950905183980088011, 7.55450369771688035071714991592